| L(s) = 1 | + 1.57·2-s + 1.09·3-s + 1.49·4-s + 1.72·6-s + 0.774·8-s + 0.196·9-s + 11-s + 1.63·12-s − 0.267·16-s − 1.35·17-s + 0.310·18-s + 1.57·22-s − 0.165·23-s + 0.847·24-s + 25-s − 0.878·27-s − 1.19·32-s + 1.09·33-s − 2.13·34-s + 0.293·36-s − 1.75·43-s + 1.49·44-s − 0.260·46-s + 1.57·47-s − 0.293·48-s + 49-s + 1.57·50-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 1.09·3-s + 1.49·4-s + 1.72·6-s + 0.774·8-s + 0.196·9-s + 11-s + 1.63·12-s − 0.267·16-s − 1.35·17-s + 0.310·18-s + 1.57·22-s − 0.165·23-s + 0.847·24-s + 25-s − 0.878·27-s − 1.19·32-s + 1.09·33-s − 2.13·34-s + 0.293·36-s − 1.75·43-s + 1.49·44-s − 0.260·46-s + 1.57·47-s − 0.293·48-s + 49-s + 1.57·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.011879596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.011879596\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 277 | \( 1 - T \) |
| good | 2 | \( 1 - 1.57T + T^{2} \) |
| 3 | \( 1 - 1.09T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.35T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 0.165T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.75T + T^{2} \) |
| 47 | \( 1 - 1.57T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.75T + T^{2} \) |
| 61 | \( 1 + 1.97T + T^{2} \) |
| 67 | \( 1 - 1.89T + T^{2} \) |
| 71 | \( 1 + 1.35T + T^{2} \) |
| 73 | \( 1 - 0.490T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.89T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.030812532467482287314728447793, −8.143870945443388876384143582141, −7.11885474501546737638148567420, −6.55167360912901415882412747982, −5.78961737209468442983055174300, −4.76955338584081087125177211016, −4.14976127257722158373765620439, −3.40190759297574068957334131295, −2.68192761684338805643139361639, −1.81928227119846924414074753857,
1.81928227119846924414074753857, 2.68192761684338805643139361639, 3.40190759297574068957334131295, 4.14976127257722158373765620439, 4.76955338584081087125177211016, 5.78961737209468442983055174300, 6.55167360912901415882412747982, 7.11885474501546737638148567420, 8.143870945443388876384143582141, 9.030812532467482287314728447793
